Robust Line Planning in case of Multiple Pools and Disruptions

0.18 MB·English

by  Apostolos Bessas

Robust Line Planning in case of Multiple Pools and
Disruptions
Apostolos Bessas1,3, Spyros Kontogiannis1,2, and Christos Zaroliagis1,3
1 R.A.Computer Technology Institute, N.Kazantzaki Str., Patras UniversityCampus,
1 26500 Patras, Greece
1 2 Computer Science Department,University of Ioannina, 45110 Ioannina, Greece
0 3 Department of Computer Engineering and Informatics, University of Patras,
2
26500 Patras, Greece

n Email: [email protected], [email protected], [email protected]
a
J January 17, 2011

4
1 Abstract. We consider the line planning problem in public transportation,
underarobustnessperspective.Wepresentamechanismforrobustlineplanning

] inthecaseofmultiplelinepools,whenthelineoperatorshaveadifferentutility
T
functionperpool.Weconductanexperimentalstudyofourmechanismonboth
G
syntheticand real-world datathat shows fast convergencetotheoptimum.We
s. alsoexploreawiderangeofscenarios,varyingfromanarbitraryinitialstate(to
c besolved)tosmalldisruptionsinapreviouslyoptimalsolution(toberecovered).
[ Ourexperimentswiththelatterscenarioshowthatourmechanismcanbeused

as an onlinerecovery schemecausing thesystem tore-convergeto itsoptimum
1
v extremely fast.
0
7
7 1 Introduction
2
.
1 Line planning is an important phase in the hierarchicalplanning process of every rail-
0 way(orpublic transportation)network4.The goalis to determine the routes(or lines)
1 of trains that will serve the customers along with the frequency each train will serve
1
a particular route. Typically, the final set of lines is chosen by a (predefined) set of
:
v candidate lines, called the line pool. In certain cases, there may be multiple line pools
i
X representing the availability of the network infrastructure at different time slots or
zones. This is due to variations in customer traffic (e.g., rush-hour pool, late evening
r
a pool, night pool), maintenance (some part of the network at a specific time zone may
be unavailable), dependencies between lines (e.g., the choice of a high-speed line may
affect the choice of lines for other trains), etc.
The line planning problem has been extensively studied under cost-oriented or
customer-oriented approaches (see e.g., [2,3,6,8]). Recently, robustness issues have
been started to be investigated. In the robust line planning problem, the task is to
provide a set of lines along with their frequencies, which are robust to fluctuations of
inputparameters;typicalfluctuationsinclude, forinstance,disruptions todailyopera-
tions (e.g., delays),or varyingcustomer demands.In [7],a game-theoreticapproachto
robust line planning was presented that delivers lines and frequencies that are robust
to delays.
Adifferentperspectiveofrobustlineplanningwasinvestigatedin[1].Thisperspec-
tive stems form recent regulations in the European Union that introduce competition
and free railway markets for its members. Under these rules the following scenario
4 For the sake of convenience, we concentrate in this work on railway networks, but the
methods and ideas developed can beapplied to any publictransportation network.
2 A. Bessas, S.Kontogiannis, and C. Zaroliagis
emerges: there is a (usually state) authority that manages the railway network infras-
tructure,referredtoastheNetwork Operator (NOP),anda(potentially)largenumber
of Line Operators (LOPs) operating as commercial organizations which want to of-
fer services to their customers using the given railway network. These LOPs act as
competing agents for the exploitation of the shared infrastructure and are unwilling
to disclose their utility functions that demonstrate their true incentives. The network
operator wishes to set up a fair cost sharing scheme for the usage of the shared re-
sources and to ensure the maximum possible level of satisfaction of the competing
agents(bymaximizingtheiraggregateutilityfunctions).Theformerimpliesaresource
pricing scheme that is robust against changes in the demands of the LOPs, while the
latter establishes a notion of a socially optimal solution, which could also be consid-
ered as a fair solution, in the sense that the averagelevel of satisfaction is maximized.
In other words, the NOP wishes to establish an incentive-compatible mechanism that
provides robustness to the system in the sense that it tolerates the agents’ unknown
incentives and elasticity of demand requests and it eventually stabilizes the system at
an equilibrium point that is as close as possible to the social optimum.
The first such mechanism, for robust line planning in the aforementioned scenario,
was presented in [1]. In that paper, the following mechanism was investigated (mo-
tivated by the pioneering work of Kelly et al. [4,5] in communication networks): the
LOPs offer bids, which they (dynamically) update for buying frequencies. The NOP
announces an (anonymous) resource pricing scheme, which indirectly implies an allo-
cation of frequencies to the LOPs, given their own bids. For the case of a single pool
of lines, a distributed, dynamic, LOP bidding and (resource) price updating scheme
was presented, whose equilibrium point is the unknown social optimum – assuming
strict concavity and monotonicity of the private (unknown) utility functions. This de-
velopment was complemented by an experimental study on a discrete variant of the
distributed, dynamic scheme on both synthetic and real-world data showing that the
mechanism converges really fast to the social optimum. The approach to the single
pool was extended to derive an analogous mechanism for the case of multiple line
pools, where it was assumed that (i) the NOP can periodically exploit a whole set of
(disjointly operating) line pools and he decides on how to divide the whole infrastruc-
ture among the different pools so that the resourcecapacity constraintsare preserved;
(ii) each LOP may be interested in different lines from different pools; and (iii) each
LOP has a single utility function which depends on the aggregate frequency that she
gets from all the pools in which she is involved.
Theaforementionedtheoreticalframeworkdemonstratedthepotentialofconverging
to the socialoptimum via a mechanism (dynamic system) that exploits the selfishness
of LOPs. A significant issue is the speed or rate of convergence of this mechanism.
Since there was no theoretical treatment of this issue, its lack was covered in [1] for
the single pool case via a complementary experimental study. Despite, however, the
significance of the convergence rate issue, the mechanism for the multiple pool case
was not experimentally evaluated in [1].
Forthecaseofmultiplelinepools,itisoftenmorerealistictoassumethateachLOP
has a different utility function per pool, since different pools are expected to provide
different profits (e.g., intercity versus regional lines, or rush-hour versus late-evening
lines).Moreover,itseemsmorenaturaltoassumethateachLOPhasadifferentutility
function per pool that depends on the frequency she gets for that pool, rather than a
single utility function that depends on the total frequency she gets across all pools.
In this work,we continue this line of researchby further investigating the multiple
pool case. In particular, we make the following contributions: (1) Contrary to the
Robust Line Planning in case of Multiple Pools and Disruptions 3
approach in [1], we consider the case where each LOP has a different utility function
foreachlinepoolsheisinterestedin,andshowhowtheapproachin[1]canbeextended
in order to provide a mechanism for this case, too. (2) We conduct an experimental
studyonadiscretevariantofthenewmechanismonbothsyntheticandreal-worlddata
demonstratingitsfastconvergencetothesocialoptimum.(3)Weconductanadditional
experimentalstudy,onbothsyntheticandreal-worlddata,toinvestigatetherobustness
of the system in the case of disruptions that affect the available capacity, which may
be reduced (due to temporary unavailability of part of the network), or increased (by
allowing usage of additional infrastructure during certain busy periods). In this case,
we show that the NOP can re-converge (recover) the system to the social optimum
pretty fast, starting from a previous optimal solution.
Therestofthepaperisorganizedasfollows.InSection2,wepresentthemechanism
for the case of multiple line pools, where each LOP has a different utility function per
pool.InSections3and4,wepresentthemaincontributionsofthisworkforthemultiple
linepoolcase;namely,theexperimentalstudyshowingthefastconvergenceofthenew
mechanism, and the robustness of the system in the case of disruptions, respectively.
We conclude in Section 5.
2 Multiple Line Pools: Different Utility Functions per Pool
Theexpositioninthissectionfollowsthatin[1].Althoughtheanalysisofourapproach
is based on the same methodology as that in [1], we have chosen to present all the
details for making the paper self-contained and for the better understanding of the
experimental studies.
In the line planning problem,the NOP providesthe public transportationnetwork
in the form of a directed graphG=(V,L), where V is the node set representing train
stations and important railway junctions, and L is the edge set representing direct
connections or links (of railway tracks) between elements of V. Each edge ℓ L is
∈
associated with a capacity c >0, which limits the number of trains that can use this
ℓ
edge in the period examined. A line p is a path in G.
For our purposes, we assume that there is set K of line pools, where each pool
representsadifferentset ofpossibleroutes (eachcorrespondingto a differentperiodof
the day). We envisionthe line pools to be implemented in disjoint time intervals (e.g.,
viasomesortoftimedivisionmultiplexing),andalsotoconcerndifferentcharacteristics
of the involved lines (e.g., high-speed pool, regular-speedpool, local-trains pool, rush-
hour pool, night-shift pool, etc.). The capacity of each resource (edge) refers to its
usage (number of trains) overthe whole time period we consider (e.g., a day), and if a
particular pool consumes (say) 50% of the whole infrastructure, then this implies that
for all the lines in this pool, each resource may exploit at most half of its capacity. It
is up to the NOP to determine how to split a whole operational period of the railway
infrastructure among the different pools, so that (for the whole period) the resource
capacity constraints are not violated.
There is also a set P of LOPs, who choose their lines from K. We assume that
each LOP p P is interested only in one line in each pool (we can always enforce
∈
this assumption by considering a LOP interested in more than one routes as different
LOPs distinguished by the specific route). Each line pool and the preference of LOPs
tolinesinitarerepresentedbyarouting matrix R(k) 0,1 |L|×|P|,k K.Eachrow
∈{ } ∈
R (k)correspondstoa differentedge ℓ L,andeachcolumnR (k)correspondsto
ℓ,⋆ ⋆,p
∈
a different LOP p P, showing which edges comprise her line in pool k.
∈
4 A. Bessas, S.Kontogiannis, and C. Zaroliagis
Each LOP p P acquires a frequency of trains that she wishes to route over
∈
her paths in R (k),k K, such that no edge capacity constraint is violated by
⋆,p
∈
the aggregate frequency running through it by all LOPs and pools. A utility function
U : R R determines the level of satisfaction of LOP p P in pool k K
p,k ≥0 ≥0
7→ ∈ ∈
forbeinggivenanend-to-endfrequencyx >0.Havingdifferentutility functionsper
p,k
pool,insteadofasingleutilityfunctionacrossallpools,ismoregenericandhencemore
realistic, since a LOP p can indeed have different valuations for different periods of a
day (rush-hour pool vs night-shift pool) and/or different types of trains (high-speed
poolvslocal-trainspool).Theseutilityfunctionsareassumedtobe strictlyincreasing,
strictlyconcave,non-negativerealfunctions ofthe end-to-endfrequency x allocated
p,k
to LOP p P in pool k K.
∈ ∈
The aggregate satisfaction level U of LOP p P across all pools is given by the
p
∈
sum of the individual gains she has in each pool, that is
U (x )=U (x ,...,x )= U (x ), p P
p p p p,1 p,k p,k p,k
∀ ∈
k∈K
X
where x = (x ) is the vector of frequencies that p gets for all the pools. The
p p,k k∈K
utility functions are private to the LOP; she is not willing to share them for competi-
tiveness reasons,not evenwith the NOP. This has a few implications on the necessary
approach to handle the problem.
The NOP, on the other hand, wishes to allocate to each LOP a frequency vector
xˆ = xˆ such that the cumulative satisfaction of all the LOPs is maximized,
p k∈K p,k
while respecting allthe edge capacityconstraint.To achievethis, the NOP divides the
P
wholerailwayinfrastructureto the pools,using variablesf ,k K thatdetermine the
k
∈
proportion of the total capacity of the edges that is assigned to pool k. Hence, the
NOP wishes to solve the following strictly convex optimization problem:
max U (x )= U (x )
p p p,k p,k
p∈P p∈Pk∈K
X X X
s.t. R (k) x c f , (ℓ,k) L K
ℓ,p p,k ℓ k
· ≤ · ∀ ∈ ×
p∈P (MSC-II)
X
f 1
k
≤
k∈K
X
x,f 0
≥
Clearly, the NOP cannot solve this problem directly for (at least) two reasons: (i)
the utility functions are unknown to him; (ii) the scale of the problem can be too
large (as it is typical with railway networks) so that it can be solved efficiently via a
centralized computation. The latter is particularly important when the whole system
is already at some equilibrium state and then suddenly a (small, relative to the size
of the whole problem) perturbation in the problem parameters occurs. Then, rather
thanhavingawholenewre-computationofthenewoptimalsolutionfromscratch,itis
particularly desirable that a dynamical scheme allows convergence to the new optimal
solution, starting from this worm start (of the previously optimal solution). All the
above reasons dictate searching for a different solution approach, that has to be as
decentralized as possible.
Optimal Solution. We adopt the approachin [1]to designa mechanismthat will be
runbythe NOPinordertosolvethe aboveproblem.Inparticular,ratherthanhaving
Robust Line Planning in case of Multiple Pools and Disruptions 5
the NOP directly deciding for the frequencies of all the LOPs in each pool, we first
let each LOP make her own bid for frequency in each pool. Then, the NOP consider
the solutionofaconvexprogramwhichis similar,butnotidenticalto (MSC-II),using
a set of (strictly increasing, strictly concave) pseudo-utilities. Our goal is to exploit
the rational (competitive) behavior of the LOPs, in order to assure that eventually
the optimal solution reached for this new program is identical to that of (MSC-II), as
required.
We start by describing the necessary and sufficient optimality (KKT) conditions
for (MSC-II). We study the Lagrangian function and exploit the economic interpre-
tation of the Lagrangian multipliers. Let Λ be the vector of Lagrangian multipliers
for the resource capacity constraints, and ζ the Lagrangian multiplier concerning the
constraintforthe capacityproportionsper pool.The Lagrangianfunction of(MSC-II)
is as follows:
L(x,f,Λ,ζ)=
U (x ) Λ R (k) x c f ζ f 1 =
p,k p,k ℓ,k ℓ,p p,k ℓ k k
− · · − · − " − #
p∈Pk∈K ℓ∈Lk∈K p∈P k∈K
X X X X X X
 
U (x ) x Λ R (k) + f cTΛ ζ +ζ =
p,k p,k p,k ℓ,k ℓ,p k ⋆,k
" − · !# · −
p∈Pk∈K ℓ∈L k∈K
X X X X (cid:2) (cid:3)
[U (x ) x µ (Λ)]+ f cTΛ ζ +ζ,
p,k p,k p,k p,k k ⋆,k
− · · −
p∈Pk∈K k∈K
X X X (cid:2) (cid:3)
whereµ (Λ) Λ R (k).We interpretΛ to be the per-unit-of-frequency
p,k ≡ ℓ∈L ℓ,k· ℓ,p ℓ,k
price of edge ℓ L in pool k K. Thus, µ (Λ) is the end-to-end per-unit price that
p,k
∈P ∈
LOP p has to pay for pool k.
The optimality (KKT) conditions for (MSC-II) are the following:
(p,k) P K, U′ (xˆ )=µˆ µ (Λˆ) (KKT-MSC-II.a)
∀ ∈ × p,k p,k p,k ≡ p,k
cT Λˆ Λˆ c =ζˆ, k K (KKT-MSC-II.b)
⋆,k ℓ,k ℓ
· ≡ · ∀ ∈
ℓ∈L
X
Λˆ R (k)xˆ c fˆ =0, (ℓ,k) L K (KKT-MSC-II.c)
ℓ,k ℓ,p p,k ℓ k
 −  ∀ ∈ ×
p∈P
X
 
ζˆ fˆ 1 =0 (KKT-MSC-II.d)
k
· − !
k∈K
X
R (k) xˆ c fˆ, (ℓ,k) L K (KKT-MSC-II.e)
ℓ,p p,k ℓ k
· ≤ · ∀ ∈ ×
p∈P
X
fˆ 1 (KKT-MSC-II.f)
k
≤
k∈K
X
xˆ,fˆ,Λˆ,ζˆ 0
≥
These conditionscharacterizethe optimalsolutionof (MSC-II).Fromthese equations,
one can observe for the optimal solution (xˆ,fˆ) and its accompanying vector of La-
grangianmultipliers (Λˆ,ζˆ):
1. By equation (KKT-MSC-II.b) all pools (at optimality) have the same (weighted
with the capacities of the edges) aggregateper-unit-of-frequency cost ζˆ.
6 A. Bessas, S.Kontogiannis, and C. Zaroliagis
2. By equation (KKT-MSC-II.d) either ζˆ= 0 or fˆ = 1. The former case (all
k∈K k
prices are set to zero) cannot actually happen, since otherwise all the pools would
P
have zero aggregate cost (i.e., every edge is underutilized in every pool). Then we
couldslightlyincreasethe frequenciesofallthe LOPsinallthe poolsby a positive
constantandkeepexactlythe samevectorofproportionsfˆ,whichwouldleadto a
strictly better solution (the utility functions are strictly increasing), contradicting
theoptimalityofxˆ.Hence,weconsideronlythecase,whereatleastsomeresources
havepositivecosts.Inthiscase,the condition fˆ =1musthold.Anyvector
k∈K k
f for which this condition holds is said to completely divide the infrastructure
P
among the pools.
Limited Information and Decentralized Mechanism. As already mentioned,
the NOP cannot know the utility functions. Therefore, a slightly different problem is
considered: Each LOP p P announces (non-negative) bids w 0 (one per pool),
p,k
∈ ≥
which she is committed to spend for acquiring frequencies in the pools. Consequently,
theNOPreplacestheunknownutilityfunctionswiththepseudo-utilitiesw log(x )
p,k p,k
inordertodetermineafrequencyvectorthatmaximizesthe aggregatelevelofpseudo-
satisfaction.Observethattheseusedpseudo-utilitiesarealsostrictlyincreasing,strictly
concavefunctions ofthe LOPs’frequencies.Thatmeans,thatNOPwishestosolvethe
following (strictly convex) optimization problem that is completely known to him:
max w log(x )
p,k p,k
p∈Pk∈K
X X
s.t. R (k) x c f , (ℓ,k) L K
ℓ,p p,k ℓ k
· ≤ · ∀ ∈ ×
p∈P (MNET-II)
X
f 1
k
≤
k∈K
X
x,f 0.
≥
The optimal solution vector (x¯,f¯), along with the corresponding vector of Lagrange
multipliers(Λ¯,ζ¯),mustsatisfythefollowingnecessaryandsufficientoptimality(KKT)
conditions for (MNET-II):
w
(p,k) P K, p,k =µ¯ µ (Λ¯) (KKT-MNET-II.a)
p,k p,k
∀ ∈ × x¯ ≡
p,k
cT Λ¯ Λ¯ c =ζ¯, k K (KKT-MNET-II.b)
⋆,k ℓ,k ℓ
· ≡ · ∀ ∈
ℓ∈L
X
Λ¯ R (k)x¯ c f¯ =0, (ℓ,k) L K (KKT-MNET-II.c)
ℓ,k ℓ,p p,k ℓ k
 −  ∀ ∈ ×
p∈P
X
 
ζ¯ f¯ 1 =0 (KKT-MNET-II.d)
k
· − !
k∈K
X
R (k) x¯ c f¯, (ℓ,k) L K (KKT-MNET-II.e)
ℓ,p p,k ℓ k
· ≤ · ∀ ∈ ×
p∈P
X
f¯ 1 (KKT-MNET-II.f)
k
≤
k∈K
X
x¯,f¯,Λ¯,ζ¯ 0
≥
Robust Line Planning in case of Multiple Pools and Disruptions 7
Thisproblemcanofcoursebesolvedinpolynomialtime,giventhebidoftheLOPs
w = (w ) . Equation (KKT-MNET-II.a) means that the NOP (correctly)
p,k (p,k)∈P×K
assigns frequency x¯ = wp,k, which is affordable at the (path) per-unit price µ¯ for
p,k µ¯p,k p,k
LOPpwhoiscommittedtoaffordanamountofw foracquiringfrequencyinpoolk.
p,k
Asalreadysaid,(x¯,f¯)istheoptimalsolutionforanybidvectordeclaredbytheLOPs,
and in particular it also holds for the true bid vector that the LOPs would really wish
to afford.
Observe, also, that problems (KKT-MSC-II) and (KKT-MNET-II) differ only in
equations (KKT-MSC-II.a) and (KKT-MNET-II.a). Our next step is to somehow as-
surethatthesetwoconditionscoincide.Tothisdirection,weexploittherationalbehav-
ioroftheLOPs:EachLOPwishestomaximizeherownaggregatelevelofsatisfaction,
therefore,she woulddeclare a bid vector that would actually achieve this. In what fol-
lows, we assume that the LOPs are price takers meaning that each of them considers
thepricesannouncedbytheNOPasconstants,withnohopeofaffectingthembytheir
own bid vector. This property is important in the following analysis, and is realistic
whenthereexistmanyLOPs,eachcontrollingonlynegligiblefractionsofthetotalflow
(or bidding process) in the system. We now state and prove an important theorem for
the existence of a mechanism for this problem.
Theorem 1. Given a transportation network G = (V,L), a set of line pools K and
a set P of selfish, price-taking LOPs, each having a private utility function for each
pool with parameter the frequency that is allocated to her in the particular pool, there
is a mechanism (a pair of a frequency allocation mechanism and a resource pricing
scheme) that computes in polynomial time the optimal solution of the sum of the utility
functions of the players, while respecting the capacities of the edges.
Proof. We create the following mechanism. The NOP completely divides the infras-
tructure and, depending on the bids of the LOPs, announces the optimal frequencies
x¯ for each LOP p P and the current resource (edge) prices Λ¯ , ℓ L,k K.
p,k ℓ,k
We have that, x¯ =∈wp,k. Each (selfish) LOP is interested in solvin∀g th∈e follo∈wing
p,k µ¯p,k
strictly convex optimization problem:
w
max (U (x¯ ) w )= U p,k w
p,k p,k p,k p,k p,k
k∈K − k∈K(cid:18) (cid:18)µ¯p,k(cid:19)− (cid:19) (MUSER-II)
X X
s.t. w 0, k K
p,k
≥ ∀ ∈
Thatis,shetriestomaximizetheutilityshegetsoverthefrequencyshewillbegranted
by the NOP, after subtracting the bid she has to pay for this frequency.
Due to the price taking property of LOP p, we can obtain the solution of the
(MUSER-II) by differentiating the objective function and setting every partial deriva-
tive equal to zero:
∂ w
U p,k w =0, k K
p,k p,k
∂w µ¯ − ∀ ∈
p,k k∈K(cid:18) (cid:18) p,k(cid:19) (cid:19)
X
we get
w 1
U′ p,k 1=0 U′ (x¯ )=µ¯
p,k µ¯ · µ¯ − ⇒ p,k p,k p,k
(cid:18) p,k(cid:19) p,k
However, this is the set of equations (KKT-MSC-II.a) of the KKT conditions for
the (MSC-II) problem.That is,everyLOPchoosesthe bids in awaythat matches the
8 A. Bessas, S.Kontogiannis, and C. Zaroliagis
equation(KKT-MSC-II.a). Butthis willhappen inthe optimalsolutionof(MNET-II)
as well. Therefore, the selfish choice of bids by the LOPs enforces the identification of
the optimal solution of MSC-II and MNET-II. I.e., we have managed to delegate the
problem of choosing the optimal bids in each iteration to the LOPs and, in doing so,
we have matched the first equation of the optimality conditions of MSC-II.
⊓⊔
This polynomially tractable mechanism (based on the solvability of MNET-II) is
totally centralized and rather inconvenient for a dynamically changing (over time),
large-scale railway system. The following lemma (whose proof can be found in the
Appendix) is crucial in deriving a dynamical system for solving (MSC-II).
Lemma 1. For any (fixed) vector f of capacity proportions that completely divides
the railway infrastructure among the pools, the optimal value of (MSC-II) exclusively
depends on the optimal vector Λ¯of the per-unit-of-frequency prices of the resources.
The above lemma suggests the following mechanism.
1. For every line pool k K, solve an instance of the single-pool case, using the
∈
decentralized mechanism in [1], obtaining the optimal solution (x ,Λ ).
⋆,k ⋆,k
2. The NOP calculates the cost of eachpool and sets the variable ζ(t) to the average
pool cost:
1
ζ(t)= cT Λ (t).
⋆,k
K ·
| |k∈K
X
Then,he updates the capacityproportionvectorf andassignsa largerpercentage
ofthetotalcapacitytothemost“expensive”linepools,sothattheircostdecreases.
This update is described by the following differential equations:
k K, f˙ (t)=max 0,cT Λ (t) ζ(t) . (1)
k ⋆,k
∀ ∈ { · − }
Note that, at the end, the vector f must be normalized, such that f = 1
k∈K k
(theproportionvectormustcompletelydividetheinfrastructureatalltimes).This
P
is done by dividing each f (t) by f (t).
k k∈K k
Roughly speaking, the convergencePof the above mechanism for a specific capacity
proportion vector f is guaranteed by the convergence of the single-pool algorithm.
Nevertheless, for the sake of completeness, we present the convergence proof in the
Appendix(Lemma2).Whenthe K single-poolinstancesaresolved,theNOPupdates
| |
the vectorf, sothatthe expensivepools getcheaper.The goalis thatallpoolsshould
have the same cost. When this happens, we know for the optimal solution of both
MNET-IIandMSC-II(x¯,f¯)andthe accompanyingLagrangemultipliers,(Λ¯,ζ¯),that:
– Equations(KKT-MNET-II.a)and(KKT-MSC-II.a)willmatchduetothefactthat
each LOP solves the (MUSER-II) problem.
– Equations (KKT-MNET-II.b) (and (KKT-MSC-II.b)) will be satisfied due to the
updates to vector f that take place at the end of each phase.
– Equations (KKT-MNET-II.c) (and (KKT-MSC-II.c)) will be satisfied due to the
resource price updating scheme chosen.
– Equations(KKT-MNET-II.d)(and(KKT-MSC-II.d))willbesatisfied,becausethe
NOP assures that at the end of each phase.
Hence, (x¯,f¯) is the optimal solution of both (KKT-MSC-II) and (MNET-II), and
thus the proposedmechanismsolves(MSC-II).The next theoremsummarizesthe pre-
ceding discussion.
Robust Line Planning in case of Multiple Pools and Disruptions 9
Theorem 2. The above dynamic scheme of resource pricing, LOPs’ bid updating and
capacity proportion updating assures the monotonic convergence of the (MNET-II)
problem to the optimal solution. The algorithm may start from any initial state of
resource prices, LOPs’ bids and capacity proportion vector.
3 Experimental Study of the Multiple-Line Pool Cases
Inthissectionwepresenttheexperimentalresultsforthemultiple-linepoolcasewhere
theLOPshavedifferentutilityfunctionsperpool.Wehaveimplementedadiscretever-
sionofthedecentralizedmechanism,whosepseudocodeisasfollows.
f (0)= 1 ;
k |K|
repeat
t=t+1;
for all k K do
∈
Solve an instance of the single-pool case for each line pool k;
end for
cost (t)=cT Λ (t);
k ⋆,k
·
ζ = Pk∈Kcostk;
|K|
for all k K do
f˙ (t)=∈max{0,costk(t)−ζ(t)};
k ζ(t)
f (t)=f (t 1)+0.1 f˙ (t);
k k k
− ·
end for
total f = f (t);
k∈K k
for all k K do
f (t)=P∈fk(t) ;
k totalf
end for
until equal costs(cost(t))
The algorithm was implemented in C++ using the GNU g++ compiler (version 4.4)
with the second optimization level (-O2 switch) on. Experiments were performed on
synthetic and real-worlddata.
Synthetic data consisted of grid graphs having a number of 7 nodes on the vertical
axis and a number of nodes in [120,360] along the horizontal axis; i.e., the size of the
grid graphs varied from 7 120 to 7 360. The capacity of each edge was randomly
× ×
chosenfrom[10,110).Fourlinepoolsweredefined.Ineachpool,therewerethreeLOPs,
each one interested in a different line. Those lines had the first edge ((0,3),(1,3)) in
common. The next edges of each line were randomly chosen each time.
Real-world data concern parts of the German railway network (mainly intercity
train connections), denoted as R1 (280 nodes and 354 edges) and R2 (296 nodes and
393edges). The capacities of the edges were in [8,16].The totalnumber oflines varies
from 100 up to 1000, depending on the size of the networks. For each network, we
defined four line pools. The second, third and fourth pool differed from the first in
about 10% of the lines (the new lines in each pool were randomly selected from the
available lines in each network).
Intheexperimentswemeasuredthenumberofiterationsneededtofindthecorrect
vectorf ofcapacityproportions(wedidnotconcentrateonthe solutionsofthe single-
poolcase,usedasasubroutine,sincethiscasewasinvestigatedin[1]). Weinvestigated
the following four scenarios:
S1: Up,1(xp,1)=104√xp,1 and Up,2(xp,2)=104√xp,2, p P.
∀ ∈
10 A. Bessas, S.Kontogiannis, and C. Zaroliagis
S2: Up,1(xp,1)= 43 ·104·√xp,1 and Up,2(xp,2)= 54 ·104·√xp,2, ∀p∈P.
S3: Up,1(xp,1)=104·√xp,1 and Up,2(xp,2)= 12 ·104·√xp,2, ∀p∈P.
S4: Up,1(xp,1)=104·√xp,1 and Up,2(xp,2)= 14 ·104·√xp,2, ∀p∈P.
WereportonexperimentswiththeR1networkandtwolinepoolsforallfourscenarios,
andonR2forscenariosS1andS2(similarresultsholdfortheotherscenarios).Table1
showstheresultsfor100,200and300linesperpoolforR1,andfor100to500linesper
pool for R2. For S1 (same utility functions), we observe a small number of necessary
Numberof lines S1S2 S3 S4 Numberof lines S1S2
100 9 33 127178 100 33 52
200 12 33 127178 200 26 49
300 19 29 128178 300 1 40
400 6 34
500 1 37
Table 1:Number ofupdates off fordifferent utility functions andnumber oflines per
pool (K =2) for R1 (left) and R2 (right).
| |
updates to the capacity proportion vector f, until the system reaches the optimum.
The main reason for this is the use of the same utility function for every pool by the
LOPs, because the algorithm starts with the initial values f = 1 and the optimal
k |K|
values in this case are quite close to these initial values. For the other scenarios with
different utility functions per pool (S2, S3, S4), we observe a larger number of the
updates required. We also observe that the more different the utility functions of each
LOP in the two pools are, the larger the number of updates required to reach the
optimum.
Another interesting observation in the case of the different utility functions per
line pool, is that the number of updates of f is almost equal. This is due to the fact
that the difference in utility functions acrossline pools has a more significanteffect on
the required number of updates than the difference in lines among the pools (in other
words, more steps are required to reach the optimal values due to the different utility
functions than due to the different costs of the line pools).
In conclusion, the number of updates required by our mechanism to converge (to
the optimal values of vector f) depends largelyon the exactparameters of the system
of differential equations (1).
4 Experimental Study of Disruptions in the Network
Weturnnowtoadifferentexperimentalstudy.Weassumethatthenetworkiscurrently
operating at optimality and that a few disruptions occur. These disruptions affect the
capacity of some edges. This can be due to technical problems leading to reducing
the capacity of those edges, or to increasing their capacity for a particular period to
handle increased traffic demand (e.g., during holidays, or rush hours) by “releasing”
more infrastructure.
We examine the behavior of the algorithms for the single and multiple pool cases
in such situations. We investigated three disruption scenarios: